Foundations of Geophysical Fluid Dynamics

Motivations

•Main motivations for the recent rapid development of Geophysical Fluid Dynamics (GFD) include the following

very important, challenging and multidisciplinary set of problems:

— Earth system modelling,

— Predictive understanding of climate variability (emerging new science!),

— Forecast of various natural phenomena (e.g., weather),

— Natural hazards, environmental protection, natural resources, etc.

What is GFD?

•Most of GFD is about dynamics of stratified and turbulent fluid on giant rotating sphere.

On smaller scales GFD is just the classical fluid dynamics with geophysical applications.

— Other planets and some astrophysical fluids (e.g., stars, galaxies) are also included in GFD.

• GFD combines applied math and theoretical physics.

It is about mathematical representation and physical interpretation of geophysical fluid motions.

•Mathematics of GFD is heavily computational, even relative to other branches of fluid dynamics (e.g., modelling

of the ocean circulation and atmospheric clouds are the largest computational problems in the history of science).

— This is because lab experiments (i.e., analog simulations) can properly address only tiny fraction of interesting

questions (e.g., small-scale waves, convection, microphysics).

• In geophysics theoretical advances are often GFD-based rather than experiment-based, because the real field

measurements are extremely complex, difficult, expensive and often impossible.

Let’s overview some geophysical phenomena of interest...

• Representation of fluid flows

Let’s consider a flow consisting of fluid particles.

Each particle is characterized by its position r and velocity u vectors:

dr(t)

dt=∂r(a, t)

∂t= u(r, t) , r(a, 0) = a

• Trajectory (pathline) of an individual fluid particle is “recording” of the

path of this particle over some time interval. Instantaneous direction of the

trajectory is determined by the corresponding instantaneous streamline.

• Streamlines are a family of curves that are instantaneously tangent to the

velocity vector of the flow u = (u, v, w). Streamline shows the direction a fluid element will travel in at any point in time.

For 2D and non-divergent flows the velocity streamfunction can be used to plot streamlines:

u = −∇×ψ , ψ = (0, 0, ψ) , u = (u, v, 0) =⇒ u = −∂ψ∂y

, v =∂ψ

∂x

Note, that u·∇ψ = 0, hence, velocity vector u always points along the isolines of ψ(x, y), implying that these isolines are indeed the

streamlines.

• Lagrangian framework: Point of view such that fluid is described by following fluid particles. Interpolation problem, not optimal use

of information.

• Eulerian framework: Point of view such that fluid is described at fixed positions in space. Nonlinearity problem.

GOVERNING EQUATIONS

• Complexity: These equations are sufficient for finding a solution but are too complicated to solve; they are useful only as a starting

point for GFD analysis.

• Art of modelling: Typically the governing equations are approximated analytically and, then, solved approximately (by analytical or

numerical methods); one should always keep track of all main assumptions and approximations.

• Continuity of mass: Consider a fixed infinitesimal volume of fluid (Eulerian view) and flow of mass through its surfaces.

∂ρ

∂t+∇·(ρu) = 0 or

Dρ

Dt= −ρ∇·u ;

D

Dt=

∂

∂t+ u·∇ ←− material derivative

Note: if fluid is incompressible (i.e., ρ = const), then the continuity equation is reduced to ∇·u = 0 , which is its common form.

• Material derivative operating on X gives the rate of change of X with the time following the fluid element (i.e., subject to a space-

and-time dependent velocity field). It is a link between the Eulerian (∂/∂t+u)·∇ and Lagrangian (D/Dt) descriptions of changes in

the fluid.

The way to see that the material derivative describes the rate of change of any property F (t, x, y, z) following a fluid particle is by

applying (i) the chain rule of differentiation and (ii) definition of velocity as the rate of change of particle position:

DF

Dt=∂F

∂t+∂F

∂x

∂x

∂t+∂F

∂y

∂y

∂t+∂F

∂z

∂z

∂t=∂F

∂t+ u

∂F

∂x+ v

∂F

∂y+ w

∂F

∂z=∂F

∂t+ u·∇F

• Tendency term, ∂X/∂t, represents the rate of change of X at a point which is fixed in space (and occupied by different fluid particles

at different times). Changes of X are observed by a stand-still observer.

• Advection term, u·∇X, represents changes of X due to movement with velocity u (or, flow supply of X to a fixed point).

Additional changes of X are experienced by an observer swimming with velocity u.

•Material tracer equation (evolution equation for composition): By similar argument, for any material tracer (e.g., chemicals, aerosols,

gases) concentration τ (amount per unit mass), the evolution equation is

∂(ρτ)

∂t+∇·(ρτu) = ρ S(τ) ,

where S(τ) stands for all the non-conservative sources and sinks of τ (e.g., boundary sources, molecular diffusion, reaction rate). The

tracer diffusion is generally added and represented by ∇·(κ∇τ), where κ is diffusivity (tensor) coefficient.

•Momentum equation: Let’s write the Newton’s Second Law in a fixed frame of reference, and for the infinitesimal volume of fluid

δV and some force F acting on a unit volume:

D

Dt(ρuδV ) = F δV =⇒ u

D

Dt(ρδV ) + ρδV

D

Dtu = F δV =⇒ Du

Dt=

1

ρF ,

where the first term on the lhs of the second equation is zero, because mass of the fluid element remains constant (i.e., no relativistic

effects).

• Pressure force can be thought as the one arising thermodynamically (due to internal motion of molecules) from the pressure p(x, y, z)acting on 6 faces of the infinitesimal cubic volume δV. Hence, the pressure force component in x is

Fx δV = [p(x, y, z)− p(x+ δx, y, z)] δy δz = −∂p∂x

δV =⇒ Fx = −∂p∂x

=⇒ F = −∇p

• Frictional force (due to internal motion of molecules) is typically approximated as ν∇2u, where ν is the kinematic viscosity.

• Body force Fb : most common examples are gravity and electromagnetic forces.

• Coriolis force is a pseudo-force that only appears in a rotating frame of reference with the rotation rate Ω : Fc = −2Ω×u.(i) It acts to deflect each fluid particle at right angle to its motion.

(ii) It doesn’t do work on a particle, because it is perpendicular to the particle velocity.

(iii) Think about motion of a moving ball on a rotating table. Foucault pendulum.

(iv) Physics of the Coriolis force: particle on a sphere deflects because of the conservation of angular momentum (when moving to the

smaller/larger latitudinal circle, it should be accelerated/decelerated to conserve its momentum).

(v) Watch some YouTube movies about the Coriolis force.

Let’s derive all the pseudo-forces in rotating coordinate systems. Rates of change of general vector B in the inertial (fixed) and

rotating (with Ω) frames of reference (indicated by i and r, respectively) are simply related:

[dB

dt

]

i=

[dB

dt

]

r+Ω×B

Let’s apply this relationship to r and ur and obtain

[dr

dt

]

i≡ ui = ur +Ω×r , (∗)

[dur

dt

]

i=

[dur

dt

]

r+Ω×ur . (∗∗)

However, we need acceleration of ui in the inertial frame and expressed completely in terms of ur and in the rotating frame.

Let’s (a) differentiate (∗) with respect to time, and in the inertial frame of reference; and (b) substitute [dur/dt]i from (∗∗) :

[dui

dt

]

i=

[dur

dt

]

r+Ω×ur +

dΩ

dt×r+Ω×

[dr

dt

]

i

Now, we again substitute [dr/dt]i from (∗) :

dΩ

dt= 0 =⇒

[dui

dt

]

i=

[dur

dt

]

r+ 2Ω×ur +Ω×(Ω×r)

The term disappearing due to the constant rate of rotation is the (minus) Euler force.

The last term is the (minus) centrifugal force. It acts a bit like gravity but in the opposite direction, hence, it can be incorporated in the

gravity force field and be “forgotten”.

To summarize, the (vector) momentum equation is:Du

Dt+ 2Ω×u = −1

ρ∇p+ ν∇2u+ Fb

Note, that in GFD the Coriolis force is traditionally kept on the lhs of the momentum equation.

• Equation of state ρ = ρ(p, T, τn) relates pressure p to the state variables — density ρ, temperature T, and chemical tracer

concentrations τn, where n = 1, 2, ... is the tracer index. All the state variables are related to matter; therefore, the equation of state is

a constitutive equation.

(a) Equations of state are often phenomenological and very different for different geophysical fluids, whereas so far other equations were

universal.

(b) The most important τn are humidity (i.e., water vapor concentration) in the atmosphere and salinity (i.e., concentration of diluted

salt mix) in the ocean.

(c) Equation of state brings in temperature, which has to be determined thermodynamically [not part of these lectures!] from internal

energy (i.e., energy needed to create the system), entropy (thermal energy not available for work), and chemical potentials corresponding

to τn (energy that can be available from changes of τn ).

(d) Example of equation of state (for sea water) involves empirically fitted coefficients of thermal expansion α, saline contraction β,and compressibility γ, which are all empirically determined functions of the state variables:

dρ

ρ=

1

ρ

( ∂ρ

∂T

)

S,pdT +

1

ρ

( ∂ρ

∂S

)

T,pdS +

1

ρ

(∂ρ

∂p

)

T,Sdp = −α dT + β dS + γ dp

• Thermodynamic equation is just one more way of writing the first law of thermodynamics, which is an expression of the conservation

of total energy. (Recall that the second law is about “arrow of time”: direction of processes in isolated systems is such that the entropy

only increases; in simple words, the heat doesn’t go from hot to cold objects.)

The thermodynamic equation can be written for T (i.e., DT/Dt = ... ), but in the GFD it is more convenient to write it for ρ :

Dρ

Dt− 1

cs2Dp

Dt= Q(ρ) ,

where cs is speed of sound, and Q(ρ) is source term (both concepts have complicated expressions in terms of the state variables).

To summarize, our starting point (assuming one material tracer) is the following COMPLETE SET OF EQUATIONS:

∂ρ

∂t+∇·(ρu) = 0 (1)

Du

Dt+ 2Ω×u = −1

ρ∇p + ν∇2u+ Fb (2)

ρ = ρ(p, T, τ) (3)

∂(ρτ)

∂t+∇·(ρτu) = ρ S(τ) (4)

Dρ

Dt− 1

cs2Dp

Dt= Q(ρ) (5)

(a) Momentum equation is for the flow velocity vector, hence, it can be written as 3 equations for the velocity components (scalars).

(b) We ended up with 7 equations and 7 unknowns (for only one tracer concentration): u, v, w, p, ρ, T, τ.

(c) Boundary and initial conditions: The governing equations (or their approximations) are to be solved subject to those.

• Spherical coordinates are natural for GFD: longitude λ, latitude θ and altitude r.

Material derivative for a scalar quantity φ in spherical coordinates is:

D

Dt=∂φ

∂t+

u

r cos θ

∂φ

∂λ+v

r

∂φ

∂θ+ w

∂φ

∂r,

where the flow velocity in terms of the corresponding unit vectors is:

u = iu+ jv + kw , (u, v, w) ≡(

r cos θDλ

Dt, r

Dθ

Dt,Dr

Dt

)

Vector analysis provides differential operators in spherical coordinates acting on

a field given by either scalar φ or vector B = iBλ + jBθ + kBr :

∇ ·B =1

cos θ

[ 1

r

∂Bλ

∂λ+

1

r

∂(Bθ cos θ)

∂θ+

cos θ

r2∂(r2Br)

∂r

]

,

∇φ = i1

r cos θ

∂φ

∂λ+ j

1

r

∂φ

∂θ+ k

∂φ

∂r,

∇2φ ≡ ∇·∇φ =1

r2 cos θ

[ 1

cos θ

∂2φ

∂λ2+

∂

∂θ

(

cos θ∂φ

∂θ

)

+ cos θ∂

∂r

(

r2∂φ

∂r

)]

,

∇×B =1

r2 cos θ

∣

∣

∣

∣

∣

∣

i r cos θ j r k

∂/∂λ ∂/∂θ ∂/∂rBλr cos θ Bθr Br

∣

∣

∣

∣

∣

∣

,

∇2B = ∇(∇·B)−∇×(∇×B) .

Writing down material derivative in spherical coordinates is a bit problematic, because the directions of the unit vectors i, j, k change

with changes in location of the fluid element; therefore, material derivatives of the unit vectors are not zeros. Note, that this doesn’t

happen in Cartesian coordinates.

•Material derivative in spherical coordinates:

Du

Dt=

Du

Dti+

Dv

Dtj +

Dw

Dtk + u

Di

Dt+ v

Dj

Dt+ w

Dk

Dt=

Du

Dti +

Dv

Dtj+

Dw

Dtk +Ωflow × u , (∗)

where Ωflow is rotation rate (relative to

the centre of Earth) of the unit vector

corresponding to the moving element of

the fluid flow:

Di

Dt= Ωflow × i ,

Dj

Dt= Ωflow × j ,

Dk

Dt= Ωflow × k .

Let’s find Ωflow by moving fluid particle in the direction of each unit vector and observing whether this motion generates any rotation.

It is easy to see that motion in the direction of i makes Ω||, motion in the direction of j makes Ω⊥, and motion in the direction

of k produces no rotation. Note (see left Figure), that Ω|| is a rotation around the Earth’s rotation axis, and it can be written as:

Ω|| = Ω|| (j cos θ + k sin θ). This rotation rate comes only from a zonally (i.e., along latitude) moving fluid element, and it can be

estimated as the following:

uδt = r cos θδλ → Ω|| ≡δλ

δt=

u

r cos θ=⇒ Ω|| =

u

r cos θ(j cos θ + k sin θ) = j

u

r+ k

u tan θ

r.

Note: the rotation rate vector in the perpendicular to Ω direction is aligned with i and given by

Ω⊥ = −i vr

=⇒ Ωflow = Ω⊥ +Ω|| = −iv

r+ j

u

r+ k

u tan θ

r=⇒

Di

Dt= Ωflow × i =

u

r cos θ(j sin θ − k cos θ) ,

Dj

Dt= −i u

rtan θ − k

v

r,

Dk

Dt= i

u

r+ j

v

r

(∗) =⇒ Du

Dt= i

(Du

Dt− uv tan θ

r+uw

r

)

+ j(Dv

Dt− u2 tan θ

r+vw

r

)

+ k(Dw

Dt− u2 + v2

r

)

The additional quadratic terms are called metric terms.

• Coriolis force also needs to be written in terms of the unit vectors of the spherical coordinates. The planetary rotation rate is always

orthogonal to the unit vector i (see Figure):

Ω = (0, Ωy, Ωz) = (0, Ωcos θ, Ω sin θ)

However, the Coriolis force projects on all the unit vectors:

2Ω×u =

∣

∣

∣

∣

∣

∣

i j k

0 2Ω cos θ 2Ω sin θu v w

∣

∣

∣

∣

∣

∣

= i (2Ωw cos θ − 2Ωv sin θ) + j 2Ωu sin θ − k 2Ωu cos θ .

By combining the metric and Coriolis terms, we obtain the most complicated set of the governing equations (other equations are changed

in the similar and trivial way) written in the spherical coordinates:

Du

Dt−(

2Ω +u

r cos θ

)

(v sin θ − w cos θ) = − 1

ρr cos θ

∂p

∂λ,

Dv

Dt+wv

r+(

2Ω +u

r cos θ

)

u sin θ = − 1

ρr

∂p

∂θ,

Dw

Dt− u2 + v2

r− 2Ωu cos θ = −1

ρ

∂p

∂r− g ,

∂ρ

∂t+

1

r cos θ

∂(uρ)

∂λ+

1

r cos θ

∂(vρ cos θ)

∂θ+

1

r2∂(r2wρ)

∂r= 0 .

Metric terms are relatively small on the surface of a large planet (r → R0) and, therefore, can be neglected for many process studies;

Note, that the gravity acceleration g is included; viscous term and external forces can be also trivially added.

• Local Cartesian approximation. Both for mathematical simplicity and for process studies, the governing equations can be written

locally for a plane tangent to the planetary surface. Then, the momentum equations can be written as

Du

Dt+ 2 (Ω cos θw − Ω sin θv) = −1

ρ

∂p

∂x,

Dv

Dt+ 2 (Ω sin θu) = −1

ρ

∂p

∂y,

Dw

Dt+ 2 (−Ωcos θu) = −1

ρ

∂p

∂z− g .

(i) Neglect Coriolis force in the vertical momentum equation, because its effect (upward/downward deflection of fluid particles, also

known as Eotvos effect), is small.

(ii) Neglect vertical velocity in the zonal momentum equation, because the corresponding component of the Coriolis force is small

relative to the other one (vertical velocity components are often small relative to the horizontal ones).

Let’s introduce the Coriolis parameter f ≡ 2Ωz = 2Ω sin θ, which is a simple nonlinear function of latitude.

(a) Theoreticians often use f -plane approximation: f = f0 (constant).

(b) Planetary sphericity is often accounted for by β-plane approximation: f(y) = f0 + βy.

The resulting equations are:Du

Dt− fv = −1

ρ

∂p

∂x,

Dv

Dt+ fu = −1

ρ

∂p

∂y,

Dw

Dt= −1

ρ

∂p

∂z− g , Dρ

Dt+ ρ∇u = 0

These equations are to be combined with the other equations (thermodynamics, material tracer, etc.) written in the local Cartesian

approximation, and even this system of equations is too difficult to solve. In order to simplify it further, we have to focus on specific

classes of fluid motions. Our main focus will be on stratified incompressible flows.

• Stratification. Let’s think about density fields in terms of their dynamic anomalies due to fluid motion and pre-existing static fields:

ρ(t, x, y, z) = ρ0 + ρ(z) + ρ′(t, x, y, z) = ρs(z) + ρ′(t, x, y, z)

Later on, the static distribution of density will be represented in terms of stacked isopycnal (i.e., constant-density) and relatively thin

fluid layers, and the dynamic density anomalies will be described by the deformations of these layers.

The pressure field can be also treated in terms of static and dynamic components:

p(t, x, y, z) = ps(z) + p′(t, x, y, z) .

We will use symbols [δρ′] and [δp′] to describe the corresponding dynamic scales.

With this concept of fluid stratification, we are ready to make one more important approximation that will affect both thermodynamic

and vertical momentum equations...

• Boussinesq approximation. It is used routinely for oceans and sometimes for atmospheres and invokes the following assumptions:

(1) Fluid incompressibility: cs=∞,(2) Small variations of static density: ρ(z)≪ ρ0 =⇒ only ρ(z) is neglected but not its vertical derivative.

(3) Anelastic approximation (used for atmospheres) is when ρ(z) is not neglected.

The thermodynamic equation in Boussinesq case (Dρ/Dt = Qρ) is traditionally written for buoyancy anomaly b(ρ) ≡ −gρ′/ρ0 :

(∗) D(b+ b)

Dt= Qb , where Qb is source term proportional to Q(ρ), and static buoyancy is b(z) ≡ −gρ/ρ0 .

Equation (∗) is often written asDb

Dt+N2(z)w = Qb , N2(z) ≡ db

dz(∗∗)

Buoyancy frequency N measures strength of the static (background) stratification in terms of its vertical derivative, in accord with (2).

Vertical momentum equation in the Boussinesq form is often written only for pressure anomaly (without the static part):

p = ps + p′ , ρ = ρs + ρ′ , −∂ps∂z

= ρsg (static balance) ,Dw

Dt= −1

ρ

∂p

∂z− g (momentum)

Let’s keep the static part for a while and rewrite the last equation in the Boussinesq approximation:

=⇒ (ρs + ρ′)Dw

Dt= −∂(ps + p′)

∂z− (ρs + ρ′) g =⇒ ρ0

Dw

Dt= −∂p

′

∂z− ρ′ g =⇒ Dw

Dt= − 1

ρ0

∂p′

∂z+ b

Note, that in the vertical acceleration term ρs + ρ′ is replaced by ρ0, in accord with (2). Horizontal momentum equations are treated

similarly.

To summarize the Boussinesq system of equations is (let’s drop primes, from now on, keeping in mind that p indicates dynamic pressure

anomaly):

Du

Dt− fv = − 1

ρ0

∂p

∂x,

Dv

Dt+ fu = − 1

ρ0

∂p

∂y,

Dw

Dt= − 1

ρ0

∂p

∂z+ b ,

∂u

∂x+∂v

∂y+∂w

∂z= 0 ,

Db

Dt+N2w = Qb

• Hydrostatic approximation. For many fluid flows vertical acceleration is small relative to gravity, and gravity force is balanced by

the vertical component of pressure gradient (we’ll come back to this approximation more formally):

Dw

Dt= −1

ρ

∂p

∂z− g =⇒ ∂p

∂z= −ρg

Hydrostatic Boussinesq approximation is commonly used for many GFD phenomena.

• Buoyancy frequency N(z) appearing in the continuous stratification case has simple physical meaning. In a stratified fluid con-

sider density difference δρ between a fluid particle adiabatically lifted by δz and surrounding fluid ρs(z). Motion of the particle is

determined by the buoyancy (Archimedes) force F and Newton’s second law:

δρ = ρparticle − ρs(z + δz) = ρs(z)− ρs(z + δz) = −∂ρs∂z

δz → F = −g δρ = g∂ρs∂z

δz

→ ρs∂2δz

∂t2= g

∂ρs∂z

δz → δz +N2δz = 0

If N2 > 0, then fluid is statically stable, and the particle will oscillate around its resting position with frequency N(z) (typical periods

of oscillations are 10− 100 minutes in the ocean, and about 10 times shorter in the atmosphere).

• Rotation-dominated flows are in the focus of GFD. Such flows are slow, in the sense that they have advective time scales longer than

the planetary rotation period: L/U ≫ f−1.Given typical observed flow speeds in the atmosphere (Ua ∼ 1−10 m/s) and ocean (Uo ∼ 0.1Ua), the length scales of rotation-

dominated flows are La ≫ 10−100 km and Lo ≫ 1−10 km. Motions on these scales constitute most of the weather and strongly

influence climate and climate variability.

Rotation-dominated flows tend to be hydrostatic (shown later).

• Thin-layered framework describes fluid in terms of stacked, vertically thin but horizontally vast layers of fluid with slightly different

densities (lighter towards the top, and heavier towards the bottom) — this is rather typical situation in GFD.

Let’s introduce the physical scales: L and H are the horizontal and vertical length scales such that L ≫ H ; and U and W are the

horizontal and vertical velocity scales, respectively, such that U ≫W. From now on, we’ll focus mostly on motion with such scales.

Thin-layered flows tend to be hydrostatic (shown later).

Summary. We considered the following sequence of simplified approximations:

Governing Equations (spherical coordinates) → Local Cartesian → Boussinesq → Hydrostatic Boussinesq.

Paid price for local Cartesian: simplified rotation and sphericity effects; neglected Ωy.

Paid price for Boussinesq: incompressible, weakly stratified (i.e., static and dynamic densities); filtered motions include acoustics,

shocks, bubbles, surface tension, inner Jupiter.

Paid price for Hydrostatic: small vertical accelerations; filtered motions include convection, breaking gravity waves, Kelvin-Helmholtz,

density currents, double diffusion, tornadoes.

Let’s consider the simplest relevant thin-layered model, which is locally Cartesian, Boussinesq and hydrostatic, and try to focus on its

rotation-dominated flow component...

BALANCED DYNAMICS

• Shallow-water model — our starting point — describes

motion of a horizontal fluid layer with variable thickness,

h(t, x, y). Density is a constant ρ0 and vertical acceleration

is neglected (hydrostatic approximation), hence:

∂p

∂z= −ρ0g → p(t, x, y, z) = ρ0g [h(t, x, y)− z] ,

where we took into account that p = 0 at z = h(t, x, y).Note, that horizontal pressure gradient is independent of z; hence,

u and v are also independent of z, and fluid moves in columns.

In local Cartesian coordinates:

Du

Dt− fv = − 1

ρ0

∂p

∂x= −g ∂h

∂x,

Dv

Dt+ fu = − 1

ρ0

∂p

∂y= −g ∂h

∂y,

whereD

Dt=

∂

∂t+ u

∂

∂x+ v

∂

∂y

Continuity equation is needed to close the system, but let’s note that vertical

velocity component is related to the height of fluid column and derive the

shallow-water continuity equation from the first principles. Recall that

velocity does not depend on z and consider mass budget of a fluid column.

The horizontal mass convergence (see earlier derivation of the continuity

equation) into the column is (apply divergence theorem):

M = −∫

S

ρ0 u·dS = −∮

ρ0hu·n dl = −∫

A

∇·(ρ0hu) dA ,

and this must be balanced by the local increase of the mass due to increase

in height of fluid column:

M =d

dt

∫

ρ0 dV =d

dt

∫

A

ρ0h dA =

∫

A

ρ0∂h

∂tdA =⇒ ∂h

∂t= −∇·(hu) =⇒ Dh

Dt+ h∇·u = 0

Note that the shallow-water continuity equation can be obtained by transformation ρ→ h.

Relative vorticity is ζ =[

∇×u]

z=∂v

∂x− ∂u

∂y; ζ > 0 is counterclockwise (cyclonic) motion, and ζ < 0 is the opposite.

Vorticity equation is obtained from the momentum equations, by taking y-derivative of the first equation and subtracting it from the

x-derivative of the second equation (remember to differentiate advection term of the material derivative; pressure terms cancel out):

Dζ

Dt+[∂u

∂x+∂v

∂y

]

(ζ + f) + vdf

dy= 0

By using the shallow-water continuity equation we obtain:

Dζ

Dt− 1

h(ζ + f)

Dh

Dt+ v

df

dy= 0 =⇒ 1

h

D(ζ + f)

Dt− 1

h2(ζ + f)

Dh

Dt= 0 =⇒ D

Dt

[ζ + f

h

]

= 0 .

• Potential vorticity (PV) material conservation law:Dq

Dt= 0 , q ≡ ζ + f

h

(a) This is a very powerful statement that reduces dynamical description of fluid motion to solving for evolution of materially conserved

scalar quantity (analogy with electric charge).

(b) PV is controlled by changes in ζ, f(y), and h (stretching/squeezing of vortex tube).

(c) Under certain conditions (e.g., when the flow is rotation-dominated) the flow can be determined entirely from PV (e.g., when

h = H = const).

(d) The above analyses can be extended to many layers and continuous stratification.

• Rossby number is ratio of the scalings for material derivative (i.e., horizontal acceleration) and Coriolis forcing:

ǫ =U2/L

fU=

U

fL

For rotation-dominated motions: ǫ≪ 1 .More conventional notation for Rossby number is Ro, but we’ll use ǫ to emphasize its smallness and apply the ǫ-asymptotic expansion.

Using the smallness of ǫ, we can expand the governing equations in terms of the geostrophic (leading-order terms) and ageostrophic

(first-order correction) motions:

u = ug + ǫua , p′ = p′g + ǫ p′a , ρ′ = ρ′g + ǫ ρ′a .

Rossby number expansion: The goal is to be able to predict strong geostrophic motions, and this requires taking into account weak

ageostrophic motions.

Let’s focus on the β-plane and mesoscales : T =L

U=

L

ǫf0L=

1

ǫf0, L/R0 ∼ ǫ =⇒ [βy] ∼ f0

R0L ∼ ǫf0 .

Let’s put the ǫ-expansion in the horizontal momentum equations and see that only pressure gradient can balance Coriolis force:

DugDt− f0 (vg + ǫva)− βy vg + ǫ2[...] = − 1

ρ0

∂pg∂x

− ǫ

ρ0

∂pa∂x

DvgDt

+ f0 (ug + ǫua) + βy ug + ǫ2[...] = − 1

ρ0

∂pg∂y

− ǫ

ρ0

∂pa∂y

ǫf0U f0U ǫf0U ǫ2f0U [p′]/(ρ0L) ǫ [p′]/(ρ0L)

• Geostrophic balance is obtained from the

horizontal momentum equations at the leading

order:

f0vg =1

ρ0

∂pg∂x

, f0ug = −1

ρ0

∂pg∂y

(a) Proper scaling for pressure must be

[p′] ∼ ρ0f0UL .

(b) Counterintuitive dynamics: Induced local

pressure anomaly results in a circular flow around it, rather than in a classical fluid flow response along the pressure gradient.

(c) It follows from the geostrophic balance, that ug is nondivergent:∂ug∂x

+∂vg∂y

= 0 (below it is shown that wg = 0 ).

(d) Geostrophic flow is 2D and nondivergent, hence, it can be described by a velocity streamfunction; note, that pressure in the

geostrophic balance acts as the streamfunction in disguise!

(e) Geostrophic balance is not a prognostic equation; the next order of the ǫ-expansion is needed to determine the flow evolution.

Now, let’s prove that geostrophically balanced flows are also hydrostatically balanced...

• Hydrostatic balance. Vertical acceleration is typically small for large-scale geophysical motions, because they are thin-layered and

rotation-dominated. Let’s prove this formally:

Dw

Dt= − 1

ρs + ρg

∂(ps + pg)

∂z− g , Dw

Dt∼ 0 ,

∂ps∂z

= −ρsg =⇒ ∂pg∂z

= −ρgg (∗)

Use the corresponding scalings W =UH/L, T =L/U, [p′]=ρ0f0UL, U = ǫf0L to identify the validity bound for the leading-

order hydrostatic balance:

Dw

Dt≪ 1

ρ0

∂pg∂z

=⇒ HU2

L2≪ ρ0f0UL

ρ0H=⇒ ǫ

(H

L

)2

≪ 1

If the last inequality is true, then vertical acceleration can be neglected — this situation routinely happens for large-scale geophysical

flows.

• Scaling for geostrophic-flow density anomaly. From the hydrostatic balance (∗) and geostrophic scaling for pressure [p′], we find

scaling for geostrophic dynamic density anomaly ρg :

[ρg] ≡ [ρ′] ∼ [p′]

gH=ρ0f0UL

gH= ρ0 ǫ

f 20L

2

gH= ρ0 ǫ F , F ≡ f 2

0L2

gH=

( L

Ld

)2

, Ld ≡√gH

f0∼ O(104 km) ,

where Ld is the external deformation scale, and F is Froude number (ratio of characteristic flow velocity to the fastest wave speed).

For many geophysical scales of interest: F ≪ 1, therefore, it is safe to assume that

F ∼ ǫ =⇒ [ρg] = ρ0 ǫ2

Thus, ubiquitous and powerful, double-balanced (geostrophic and hydrostatic) motions correspond to nearly flat isopycnals.

• Continuity for ageostrophic flow. Let’s now turn attention to the continuity equation and also ǫ-expand it:

∂ρ

∂t+∂(ρu)

∂x+∂(ρv)

∂y+∂(ρw)

∂z= 0 , ρ = ρs + ρg, u = ug + ǫ ua, v = vg + ǫ va, w = wg + ǫ wa →

∂ρg∂t

+ (ρs + ρg)(∂ug∂x

+∂vg∂y

)

+ ug∂ρg∂x

+ vg∂ρg∂y

+ ǫρs

(∂ua∂x

+∂va∂y

)

+ ǫ2 [...] +∂

∂z(wgρs + ǫwaρs + wgρg + ǫwaρg) = 0

Use∂ug∂x

+∂vg∂y

= 0 and ρg ∼ ǫ2 to obtain at the leading order:∂(wgρs)

∂z= 0 −→ wgρs = const

Because of the BCs, somewhere in the water column wg(z) has to be zero =⇒ wg = 0 , w = ǫ wa, [w] =W = ǫ UH

L

At the next order of the ǫ-expansion we recover the continuity equation for ageostrophic flow component:

∂(waρs)

∂z+ ρs

(∂ua∂x

+∂va∂y

)

= 0 .

Let’s keep this in mind and use it in the derivation of geostrophic vorticity equation.

• Geostrophic (absolute) vorticity equation is obtained by going to the next order of ǫ in the shallow-water momentum equations:

DgugDt

− (ǫf0va + vgβy) = −ǫ1

ρs

∂pa∂x

,DgvgDt

+ (ǫf0ua + ugβy) = −ǫ1

ρs

∂pa∂y

,Dg

Dt≡ ∂

∂t+ ug

∂

∂x+ vg

∂

∂y.

(i) Take curl of the above equations (i.e., subtract y-derivative of the first equation from x-derivative of the second equation) and mind

complexity of the material derivative;

(ii) Use nondivergence of the geostrophic velocity;

(iii) Use continuity equation for ageostrophic flow to replace horizontal ageostrophic velocity divergence.

Thus, we obtain the geostrophic vorticity equation:

DgζgDt

+ βvg =Dg

Dt[ζg + βy] = ǫ

f0ρs

∂(ρswa)

∂z, ζg ≡

∂vg∂x− ∂ug

∂y

(a) This looks almost as PV material conservation law, but unfortunately it is not one, because of the rhs term. Can it be reformulated in

the missing PV component?

(b) Evolution of the absolute vorticity is determined by divergence of the vertical mass flux, due to tiny vertical velocity. This is the

process of squeezing or stretching the isopycnals, which is related to the form drag mechanism discussed below. Can this term be

determined from the leading-order fields?

(c) Quasigeostrophic theory expresses the rhs in terms of vertical movement of isopycnals, then, it relates this movement to pressure,

which is the streamfunction in disguise.

(d) On the other hand, evolution of the horizontal absolute vorticity produces the squeezing or stretching deformations, which induce

motions in the neighbouring layers.

(e) If ρs is constant within the layer (i.e., thin-layered framework), then, it cancels out from the rhs, and we are left with the vertical

component of velocity divergence.

QUASIGEOSTROPHIC THEORY

• Two-layer shallow-water model is a natural extension of the

single-layer shallow-water model. It illuminates effects of isopycnal

deformations on the geostrophic vorticity. This model can be

straightforwardly extended to many isopycnal (i.e., constant-density)

layers, thus, producing the family of isopycnal models.

The model assumes geostrophic and hydrostatic balances, and usual

Boussinesq treatment of density:

∆ρ ≡ ρ2 − ρ1 ≪ ρ1, ρ2 , ρ1 ≈ ρ2 ≈ ρ .

All notations are introduced on the sketch.

The layer thicknesses and pressures consist of the static and dynamic

components:

h1(t, x, y) = H1 +H2 + η1(t, x, y) , h2(t, x, y) = H2 + η2(t, x, y) ,

p1 = ρ1g(H1 +H2 − z) + p′1(t, x, y) , p2 = ρ1gH1 + ρ2g(H2 − z) + p′2(t, x, y) ,

Here, the shallow-water dynamic pressure anomalies are independent of z, as we have seen, and the static pressures are obtained by

integrating the hydrostatic balance ∂p/∂z=−ρg, and taking the integration constants such that the static pressure is zero at the upper

surface and continuous on the internal interface between the layers.

• Continuity boundary condition for pressure is just a component of the continuity boundary condition for stress tensor (sometimes, this

involves surface tension); here, it allows to relate dynamic pressure anomalies and isopycnal deformations.

In the two-layer model this boundary condition is equivalent to saying that:

(a) pressure at the upper surface must be zero (more generally, it must be equal to the atmospheric pressure),

(b) pressure on the internal interface must be continuous, i.e., p1 = p2 = P.

Note, that in the absence of motion (p′1 = p′2 = 0) both of these conditions are automatically satisfied for the static pressure component:

p1|z=H1+H2= 0 , p1|z=H2

= p2|z=H2= ρ1gH1 .

In the presence of motion, the upper-surface pressure continuity statement p1|z=η1+H1+H2= 0 translates into

p′1(t, x, y) = ρ1gη1(t, x, y) .

On the internal interface, the pressure continuity statement is:

P = p1|z=η2+H2= ρ1g(H1−η2)+p′1 , P = p2|z=η2+H2

= ρ1gH1−ρ2gη2+p′2 =⇒ p′2(t, x, y) = p′1(t, x, y)+g∆ρ η2(t, x, y)

Thus, by using expression for the upper-layer pressure, we obtain:

p′2(t, x, y) = ρ1gη1(t, x, y) + g∆ρ η2(t, x, y)

• Geostrophy at the leading order links horizontal velocities and slopes of the isopycnals (interfaces) in the upper and deep layers:

−f0v1 = −g∂η1∂x

, f0u1 = −g∂η1∂y

; −f0v2 = −gρ1ρ2

∂η1∂x− g ∆ρ

ρ2

∂η2∂x

, f0u2 = −gρ1ρ2

∂η1∂y− g ∆ρ

ρ2

∂η2∂y

Next, we recall that ρ1 ≈ ρ2 ≈ ρ (Boussinesq argument), introduce the reduced gravity g′ ≡ g∆ρ/ρ, and, thus, simplify the second-

layer equations:

−f0v2 = −g∂η1∂x− g′ ∂η2

∂x, f0u2 = −g

∂η1∂y− g′ ∂η2

∂y

• Geostrophic vorticity equations.

Now, let’s take a look at the full system of the two-layer shallow-water equations:

Du1Dt− fv1 = −g

∂η1∂x

,Dv1Dt

+ fu1 = −g∂η1∂y

,∂(h1 − h2)

∂t+∇·((h1 − h2)u1) = 0 ,

Du2Dt− fv2 = −g

∂η1∂x− g′∂η2

∂x,

Dv2Dt

+ fu2 = −g∂η1∂y− g′∂η2

∂y,

∂h2∂t

+∇·(h2u2) = 0 .

As we have argued, at the leading order the momentum equations are geostrophic; at the ǫ-order, we can formulate the layer-wise

vorticity equations with the additional rhs terms responsible for vertical deformations. For this purpose:

(a) Expand the momentum equations in terms of ǫ,(b) take curl of the momentum equations (∂(2)/∂x− ∂(1)/∂y),(c) replace divergence of the horizontal ageostrophic velocity (ua, va) with the vertical divergence of wa.The resulting equations are:

DnζnDt

+ βvn = f0∂wn

∂z,

Dn

Dt=

∂

∂t+ un

∂

∂x+ vn

∂

∂y, ζn ≡

∂vn∂x− ∂un

∂y, n = 1, 2

Within each layer horizontal velocity does not depend on z, therefore, vertical integrations of the vorticity equations across each layer

yield (here, we assume nearly flat isopycnals everywhere by replacing h1 − h2 ≈ H1 and h2 ≈ H2 on the lhs):

H1

(D1ζ1Dt

+ βv1

)

= f0(

w1(h1)− w1(h2))

, H2

(D2ζ2Dt

+ βv2

)

= f0 w2(h2) , (∗)

Thus, we extended the assumption of nearly flat isopycnals to everywhere, beyond the scale of motions. Note, that in (*) we took

w2(bottom) = 0, but this is true only for the flat bottom (along topographic slopes vertical velocity can be non-zero, as only normal-to-

boundary velocity component vanishes).

• Vertical movement of isopycnals in terms of pressure can be obtained, and this step closes the equations. For that, we use kinematic

boundary condition, which comes from considering fluid elements on the fluid interface or surface, such that the vertical coordinates of

these elements are given by z = h(t, x, y). Next, let’s consider functional F (t, x, y, z) = h(t, x, y) − z, and acknowledge, that it is

always zero for a fluid elements sitting on the interface or surface; hence, its material derivative is zero:

DF

Dt= 0 =

Dh

Dt− w ∂z

∂z→ w =

Dh

Dt

By combining the kinematic boundary condition with the Boussinesq argument (ρ1≈ρ2≈ρ), we obtain:

wn(hn) =DnhnDt

=DnηnDt

=⇒ w1(h1) =1

ρg

D1p′1

Dt, w1,2(h2) =

1

∆ρg

D1,2(p′2 − p′1)Dt

(∗∗)

• Pressure is streamfunction in disguise.

In each layer geostrophic velocity streamfunction is linearly related to dynamic pressure anomaly, as follows from the geostrophic

momentum balance:

f0vn =1

ρ

∂p′n∂x

, f0un = −1

ρ

∂p′n∂y

=⇒ ψn =1

f0ρp′n , un = −∂ψn

∂y, vn =

∂ψn

∂x(∗ ∗ ∗)

Relative vorticity ζ is always conveniently expressed in terms of ψ : ζ =∂v

∂x− ∂u

∂y= ∇2ψ

Let’s now combine (∗), (∗∗) and (∗ ∗ ∗) to obtain the fully closed equations predicting evolution of the leading-order streamfunction.

• Two-layer quasigeostrophic (QG) model.

D1ζ1Dt

+ βv1 −f 20

gH1

( ρ

∆ρ

D1

Dt(ψ1 − ψ2) +

D1ψ1

Dt

)

= 0 ,

D2ζ2Dt

+ βv2 − f 20

gH2

ρ

∆ρ

D2

Dt(ψ2 − ψ1) = 0

(a) Note that ∆ρ ≪ ρ, therefore the last term of the first equation is neglected (i.e., the rigid-lid approximation is taken; it states that

the surface elevation is much smaller than the internal interface displacement).

(b) Familiar reduced gravity is g′ ≡ g∆ρ/ρ, and stratification parameters are defined as S1 =f 20

g′H1

, S2 =f 20

g′H2

.

(c) Dimensionally, [S1] ∼ [S2] ∼ L−2 → QG (i.e., a double-balanced) motion of a stratified fluid operates on the internal defor-

mation scales: R1 = 1/√S1, and R2 = 1/

√S2, which are O(100km) in the ocean and about 10 times larger in the atmosphere

(fundamentally, (Rn ≪ Ld=f20 /gH, because g′ ≪ g) .

With the above information taken into account, we obtain the final set of the QG equations:

D1

Dt

[

∇2ψ1 − S1 (ψ1 − ψ2)]

+ βv1 = 0 ,D2

Dt

[

∇2ψ2 − S2 (ψ2 − ψ1)]

+ βv2 = 0

Potential vorticity anomalies are defined as q1 = ∇2ψ1 − S1 (ψ1 − ψ2), q2 = ∇2ψ2 − S2 (ψ2 − ψ1)

Note: These expressions for the PV anomalies can be obtained by linearization of the full shallow-water PV (given without proof).

• Potential vorticity (PV) material conservation law.

(Absolute) PV is defined as Π1 = q1 + f = q1 + f0 + βy, Π2 = q2 + f = q2 + f0 + βy .

(a) PV is materially conserved quantity:Dn

DtΠn =

∂Πn

∂t+∂ψn

∂x

∂Πn

∂y− ∂ψn

∂y

∂Πn

∂x= 0 , n = 1, 2

(b) PV can be considered as a “charge” advected by the flow; but this is active charge, as it defines the flow itself.

(c) PV inversion brings in intrinsic and important spatial nonlocality of the velocity field around “elementary charge” of PV:

Π1 = ∇2ψ1 − S1 (ψ1 − ψ2) + βy + f0 , Π2 = ∇2ψ2 − S2 (ψ2 − ψ1) + βy + f0

(d) PV consists of of relative vorticity, density anomaly (resulting from isopycnal displacement), and planetary vorticity.

Other comments:

(a) Midlatitude theory: QG framework does not work at the equator, where f = 0.

(b) Vertical control: Nearly horizontal geostrophic motions are determined by vertical stratification, vertical component of ζ , and vertical

isopycnal stretching.

(c) Four main assumptions that have been made:

(i) Rossby number ǫ is small (hence, the expansion focuses on mesoscales);

(ii) β-plane approximation and small meridional variations of Coriolis parameter;

(iii) isopycnals are nearly flat ([δρ′] ∼ ǫFρ0 ∼ ǫ2ρ0) everywhere;

(iv) hydrostatic Boussinesq balance.

ROSSBY WAVES

• In the broad sense, Rossby wave is inertial wave propagating on the background PV gradient.

First discovered in the Earth’s atmosphere.

• Oceanic Rossby waves are more difficult to observe (e.g., altimetry, in situ measurements)

• Sea surface height anomalies

propagating to the west are signatures

of baroclinic Rossby waves.

• To what extent transient flow anomalies

can be characterized as waves rather

than isolated coherent vortices remains

unclear.

⇐= Visualization of oceanic eddies/waves

by virtual tracer

Flow speed from the high

resolution computation

shows many eddies/waves =⇒

•Many properties of the flow

fluctuations can be interpreted

in terms of the linear (Rossby)

waves

• General properties of waves.

(a) Waves provide interaction mechanism which is both long-range and fast relative to flow advection.

(b) Waves are observed as periodic propagating (and standing) patterns, e.g., ψ = ReA exp[i(kx+ ly+mz−ωt+φ)], characterized

by amplitude, wavenumbers, frequency, and phase.

Wavevector is defined as the ordered set of wavenumbers: K=(k, l,m).

(c) Dispersion relation comes from the dynamics and connects frequency and wavenumbers, and, thus, yields phase speeds and group

velocity Cg.

(d) Phase speeds along the axes of coordinates are rates at which intersections of the phase lines with each axis propagate along this axis:

C(x)p =

ω

k, C(y)

p =ω

l, C(z)

p =ω

m;

these speeds do not form a vector (note that phase speed along an axis increases with decreasing projection of K on this axis).

(e) The fundamental phase speed Cp = ω/K, where K = |K|, is defined along the wavevector. This is natural, because waves

described by complex exponential functions have instantaneous phase lines perpendicular to K.The fundamental phase velocity (vector) is defined as

Cp =ω

|K|K

|K| =ω

K2K .

(f) The group velocity (vector) is defined as

Cg =(∂ω

∂k,∂ω

∂l,∂ω

∂m

)

.

(g) Propagation directions: phase propagates in the direction of K; energy (hence, information!) propagates at some angle to K.

(h) If frequency ω = ω(x, y, z) is spatially inhomogeneous, then trajectory traced by the group velocity is called ray, and the path of

waves is found by ray tracing methods.

•Mechanism of Rossby waves. Consider the simplest 1.5-layer (a.k.a. the equivalent barotropic) QG PV model, which is obtained by

considering H2 →∞ in the two-layer QG PV model:

∂Π

∂t+∂ψ

∂x

∂Π

∂y− ∂ψ

∂y

∂Π

∂x= 0 , Π = ∇2ψ − 1

R2ψ + βy ,

where R−2 = S1 is the stratification parameter written in terms of the inverse length scale parameter R.By introducing the Jacobian operator J(A,B) = AxBy − AyBx, the corresponding equivalent-barotropic equation can be written as

∂

∂t

(

∇2ψ − 1

R2ψ)

+ J(

ψ,∇2ψ − 1

R2ψ)

+ β∂ψ

∂x= 0 . (∗)

Note, that in the limit R → ∞ the dynamics becomes purely 2D and deformations of the layer thickness become infinitesimal; this is

equivalent to g′ → g.

We are interested in small-amplitude flow disturbances around

the state of rest; the corresponding linearized equation (∗) is

∂

∂t

(

∇2ψ − 1

R2ψ)

+ β∂ψ

∂x= 0

→ ψ ∼ ei(kx+ly−ωt) →

−iω(

− k2 − l2 − 1

R2

)

+ iβk = 0

Thus, the resulting Rossby waves dispersion relation is ω =−βk

k2 + l2 +R−2.

Plot dispersion relation, discuss zonal, phase and group speeds...

Consider a timeline in the fluid at rest, then, perturb it (see Figure): the resulting westward propagation of Rossby waves is due to the

β-effect and material PV conservation.

•Mean-flow effects. Consider small-amplitude flow disturbances around some background flow given by its streamfunction Ψ(x, y, z).What happens with the dispersion relation and, hence, with the waves?

To simplify the problem, let’s stay with the 1.5-layer QG PV model, consider uniform, zonal background flow Ψ = −Uy, and substitute:

ψ → −Uy + ψ, Π→(

β +U

R2

)

y +∇2ψ − 1

R2ψ .

The linearized dynamics is the following:

( ∂

∂t+ U

∂

∂x

)(

∇2ψ − 1

R2ψ)

+∂ψ

∂x

(

β +U

R2

)

= 0 → ψ ∼ ei(kx+ly−ωt) → ω = kU − k (β + UR−2)

k2 + l2 +R−2

(a) In the dispersion relation, the first term kU is Doppler shift, which is due to advection of the wave by the background flow;

(b) The second term contains effect of the altered background PV;

(c) There are also corresponding changes in the group velocity;

(d) Complicated 2D and 3D background flows profoundly influence Rossby waves properties, but the corresponding dispersion relations

are difficult to obtain.

• Two-layer Rossby waves. Consider now the two-layer QG PV equations linearized around the state of rest:

∂

∂t

[

∇2ψ1 −1

R21

(ψ1 − ψ2)]

+ β∂ψ1

∂x= 0 ,

∂

∂t

[

∇2ψ2 −1

R22

(ψ2 − ψ1)]

+ β∂ψ2

∂x= 0 , R2

1 =g′H1

f 20

, R22 =

g′H2

f 20

Diagonalization of the dynamics: the governing equations can be decoupled from each other by the linear transformation of variables

from the layer-wise streamfunctions to the streamfunctions of the vertical modes.

The barotropic mode φ1 and the first baroclinic mode φ2 are defined as

φ1 ≡ ψ1H1

H1 +H2+ ψ2

H2

H1 +H2, φ2 ≡ ψ1 − ψ2 .

These modes represent separate (i.e., governed by different dispersion relations) families of Rossby waves:

∂

∂t∇2φ1 + β

∂φ1

∂x= 0 → ω1 = −

βk

k2 + l2

∂

∂t

[

∇2φ2 −1

R2D

φ2

]

+ β∂φ2

∂x= 0 , RD ≡

[ 1

R21

+1

R22

]−1/2

→ ω2 = −βk

k2 + l2 +R−2D

where RD is referred to as the first baroclinic Rossby radius.

(a) The diagonalizing layers-to-modes transformation and its inverse (modes-to-layers) transformation are linear operations. The (pure)

barotropic mode can be written in terms of layers as

ψ1 = ψ2 = φ1 ,

therefore, it is vertically uniform and actually describes the vertically averaged flow. Barotropic waves are fast (typical periods are

several days in the ocean and 10 times faster in the atmosphere); their dispersion relation does not depend on the stratification.

(b) The (pure) baroclinic mode can be written in terms of layers as

ψ1 = φ2H2

H1 +H2, ψ2 = −φ2

H1

H1 +H2→ ψ2 = −

H1

H2ψ1 .

therefore, it changes sign vertically, and its vertical integral iz sero. Baroclinic waves are slow (typical periods are several months in the

ocean and 10 times faster in the atmosphere); they can be viewed as propagating anomalies of the pycnocline (thermocline), because the

streamfunction has vertical derivative.

• Continuously stratified Rossby waves.

Continuously stratified model is a natural extension of the isopycnal

model with a large number of layers.

The corresponding linearized QG PV dynamics is given by

∂

∂t

[

∇2ψ +f 20

ρs

∂

∂z

( ρsN2(z)

∂ψ

∂z

)]

+ β∂ψ

∂x= 0

→ ψ ∼ Φ(z) ei(kx+ly−ωt) →

f 20

ρs

d

dz

( ρsN2(z)

dΦ(z)

dz

)

=(

k2 + l2 +kβ

ω

)

Φ(z) ≡ λΦ(z) (∗)

Boundary conditions at the top and bottom are to be specified, e.g., by

imposing zero density anomalies:

ρ ∼ dΦ(z)

dz

∣

∣

∣

z=0,−H= 0 . (∗∗)

Combination of (∗) and (∗∗) is an eigenvalue problem that can be solved for a discrete spectrum of eigenvalues and eigenmodes.

(a) Eigenvalues λn yield dispersion relations ωn=ωn(k, l), and the corresponding eigenmodes φn(z) are the vertical normal modes,

like the familiar barotropic and first baroclinic modes in the two-layer case.

(b) The Figure illustrates the first, second and third baroclinic modes for the oceanic stratification.

(c) The corresponding baroclinic Rossby deformation radius R(n)D ≡ λ

−1/2n characterizes horizontal length scale of the nth vertical

mode. The higher is the mode, the more oscillatory it is in vertical, and the slower it propagates.

(d) The (zeroth) barotropic mode has R(0)D =∞ and λ0 = 0.

(e) The first Rossby deformation radius R(1)D is the most important fundamental length scale of the geostrophic turbulence.